Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

TL;DR

This paper develops a new bialgebra theory for differential algebras, introducing differential antisymmetric infinitesimal (ASI) bialgebras, and explores their connections to Poisson bialgebras, derivations, and related algebraic structures.

Contribution

It generalizes ASI bialgebras to differential algebras, introduces coherent derivations, and extends Poisson bialgebra constructions within this framework.

Findings

Characterization of differential ASI bialgebras via double constructions and matched pairs.

Introduction of $ ext{O}$-operators and differential dendriform algebras for constructing these bialgebras.

Extension of Poisson algebra and bialgebra theory using differential Zinbiel algebras.

Abstract

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of $\mathcal{O}$-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.